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A representation SH* U(s) of A in H is said to be topologically irreducible
if there exists no closed vector subspace E of H, other than {0} and H, which
is stable with respect to U.

From (15.5.3) we obtain the following irreducibility criterion:

(15.5.4) In order that U should be topologically irreducible, it is necessary
and sufficient that the only orthogonal projections P such that PU(s) = U(s)P
for all s e A should be 0 and 1H .

For this condition expresses that {0} and H are the only closed subspaces
of H which are stable under U.

(15.5.5)    Let SH* U(s) be a representation of A in H, let E be the closure in H
of the vector subspace generated by the elements U(s)  x for s e A and x e H,
and let E' be the set of all x ell such that U(s) - x = 0 for all s 6 A. Then
E and E' are stable with respect to U and are orthogonal supplements of each
other in H.

Since U(sf) = U(s)U(t), it is clear that E and E' are stable subspaces of H.
Let E" be the orthogonal supplement of E in H. We have seen earlier that E"
is stable with respect to U. But if x e E", we have 17(5-)  x e E by definition,
hence U(s)  x e E n E" = {0} for all s e A, so that E" <= E'. Conversely, if
x e E', s e A and y e H, we have (x \ U(s)  y) = (U(s*)  x \ y) = 0 by defini-
tion, so that x is orthogonal to E, and therefore E' cz E".

The subspace E is called the essential subspace for U. If E' = {0}, the
representation U is said to be nondegenerate. An equivalent condition is that
the elements U(s) - x should form a total subspace of H; by (15.5.2), this is
always the case if A has a unit element.

A vector x0 e H is a totalizer or totalizing vector for a representation U of
A in H if the vector subspace of H generated by the transforms U(s)  x0 of
XQ , as s runs through A, is dense in H. (In any case, this subspace is stable with
respect to U.) A representation which admits a totalizer is said to be topologi-
cally cyclic. If U is topologically irreducible, every nonzero vector x0 e H is a
totalizer, and conversely.

(15.5.6)     Suppose that A has a unit element. Let s\-+ U(s) be a nondegenerate
representation of A in a separable Hilbert space H. Then H is the Hilbert sum
of a sequence (Hn)( finite or infinite) of closed subspaces,, stable with respect to U
and such that the restriction of U to each Hn is topologically cyclic.(s)  x =  Un(s) - xn, and